BEHAVIOR OF THE RING CLASS NUMBERS OF A REAL QUADRATIC FIELD

Let K be a real quadratic field Q(root n) with an integer n = df(2) with the field discriminant d of K and f >= 1. Q. Mushtaq found an interesting phenomena that any totally negative number kappa(0) with kappa(0) < 0 and kappa(sigma)(0) < 0 belonging to the discriminant n, attains an ambiguous number kappa(m) with kappa(m)kappa(sigma)(m) < 0 after a finitely many actions kappa(Aj)(0) with 0 <= j <= m by modular transformations A(j) is an element of SL2+(Z). Here sigma denotes the embedding of K distinct from the identity. In this paper we give a new aspect for the process to reach an ambiguous number from a totally negative or totally positive number, by which the gap of the proof of Q. Mushtaq's Theorem is complemented. Next as an analogue of Gauss' Genus Theory, we prove that the ring class number h(+)(df(2)), coincides with the ambiguous class number belonging to the discriminant n = df(2) and it's behavior is unbounded, when f with suitable prime factors goes to infinity using the ring class number formula.